In 2004, I asked Merriam-Webster the following question. Assistant Editor, Jennifer N. Cislo, replied as follows.

Q: “Suasion” means “the act of influencing or persuading.” “Persuasion” means, inter alia, “the act or process or an instance of persuading.” How do “suasion” and “persuasion” differ?

A: Apart from their spelling, there is only niggling if any difference between suasion and persuasion. The only notable difference is that persuasion is a far more commonly used word for describing any act, instance, or process of influencing another.